Theorem REF  improves on results in [1]} for the Ising model
by allowing a wider rage for \(\beta \) .
Specifically, in the ferromagnetic case the above bound allows for \(\beta \)  that is a constant factor larger
than what we had before. The analogous holds for the antiferromagnetic Ising, i.e., Theorem REF 
allows for \(\beta \)  which is a constant factor smaller than what we had before.
